Equations and Inequalities and Absolute Value worksheet | Live ... - Free Printable

Let's solve each equation and inequality step by step.

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Solve each equation

#### 1) $ 7(1 - 3r) = 91 $

Distribute:
$$
7 - 21r = 91
$$

Subtract 7 from both sides:
$$
-21r = 84
$$

Divide by -21:
$$
r = -4
$$

✔ Answer: $ r = -4 $

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#### 2) $ -320 = 8(7a - 1) - 4a $

Distribute:
$$
-320 = 56a - 8 - 4a
$$

Combine like terms:
$$
-320 = 52a - 8
$$

Add 8 to both sides:
$$
-312 = 52a
$$

Divide by 52:
$$
a = -6
$$

✔ Answer: $ a = -6 $

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#### 3) $ -6 + a = -( -7a - 6 ) $

Simplify the right side:
$$
-6 + a = 7a + 6
$$

Subtract $ a $ from both sides:
$$
-6 = 6a + 6
$$

Subtract 6:
$$
-12 = 6a
$$

Divide by 6:
$$
a = -2
$$

✔ Answer: $ a = -2 $

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#### 4) $ -4n + 5 = 5(1 + 3n) $

Distribute:
$$
-4n + 5 = 5 + 15n
$$

Subtract 5 from both sides:
$$
-4n = 15n
$$

Subtract $ 15n $:
$$
-19n = 0
$$

Divide:
$$
n = 0
$$

✔ Answer: $ n = 0 $

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#### 5) $ |6n| = 48 $

Absolute value means:
$$
6n = 48 \quad \text{or} \quad 6n = -48
$$

Solve:
$$
n = 8 \quad \text{or} \quad n = -8
$$

✔ Answer: $ n = 8 $ or $ n = -8 $

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#### 6) $ |-8v| = 64 $

Since absolute value makes it positive:
$$
|-8v| = 8|v| = 64 \Rightarrow |v| = 8
$$

So:
$$
v = 8 \quad \text{or} \quad v = -8
$$

✔ Answer: $ v = 8 $ or $ v = -8 $

---

#### 7) $ |b - 1| + 3 = 4 $

Subtract 3:
$$
|b - 1| = 1
$$

So:
$$
b - 1 = 1 \quad \text{or} \quad b - 1 = -1
$$

Solve:
$$
b = 2 \quad \text{or} \quad b = 0
$$

✔ Answer: $ b = 2 $ or $ b = 0 $

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#### 8) $ 4\left|\frac{x}{3}\right| = 4 $

Divide both sides by 4:
$$
\left|\frac{x}{3}\right| = 1
$$

So:
$$
\frac{x}{3} = 1 \quad \text{or} \quad \frac{x}{3} = -1
$$

Multiply:
$$
x = 3 \quad \text{or} \quad x = -3
$$

✔ Answer: $ x = 3 $ or $ x = -3 $

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#### 9) $ |9p + 3| = 42 $

Set up two equations:
$$
9p + 3 = 42 \quad \text{or} \quad 9p + 3 = -42
$$

First:
$$
9p = 39 \Rightarrow p = \frac{13}{3}
$$

Second:
$$
9p = -45 \Rightarrow p = -5
$$

✔ Answer: $ p = \frac{13}{3} $ or $ p = -5 $

---

#### 10) $ -9 + |7n - 6| = 55 $

Add 9:
$$
|7n - 6| = 64
$$

So:
$$
7n - 6 = 64 \quad \text{or} \quad 7n - 6 = -64
$$

First:
$$
7n = 70 \Rightarrow n = 10
$$

Second:
$$
7n = -58 \Rightarrow n = -\frac{58}{7}
$$

✔ Answer: $ n = 10 $ or $ n = -\frac{58}{7} $

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Solve each inequality

#### 11) $ -180 < -5(-4x + 8) $

Distribute:
$$
-180 < 20x - 40
$$

Add 40:
$$
-140 < 20x
$$

Divide by 20:
$$
-7 < x \quad \text{or} \quad x > -7
$$

✔ Answer: $ x > -7 $

---

#### 12) $ 7 - 3(7n - 7) \leq -140 $

Distribute:
$$
7 - 21n + 21 \leq -140
$$

Simplify:
$$
28 - 21n \leq -140
$$

Subtract 28:
$$
-21n \leq -168
$$

Divide by -21 (reverse inequality):
$$
n \geq 8
$$

✔ Answer: $ n \geq 8 $

---

#### 13) $ 23 - 4n \leq -(-2 + n) $

Simplify right side:
$$
23 - 4n \leq 2 - n
$$

Add $ 4n $ to both sides:
$$
23 \leq 2 + 3n
$$

Subtract 2:
$$
21 \leq 3n
$$

Divide by 3:
$$
7 \leq n \quad \text{or} \quad n \geq 7
$$

✔ Answer: $ n \geq 7 $

---

#### 14) $ 3 + 3(6 + 4x) \leq -24 + 3x $

Distribute:
$$
3 + 18 + 12x \leq -24 + 3x
$$

Simplify:
$$
21 + 12x \leq -24 + 3x
$$

Subtract $ 3x $:
$$
21 + 9x \leq -24
$$

Subtract 21:
$$
9x \leq -45
$$

Divide by 9:
$$
x \leq -5
$$

✔ Answer: $ x \leq -5 $

---

#### 15) $ |-3 + x| > 3 $

Rewrite as:
$$
|x - 3| > 3
$$

This means:
$$
x - 3 > 3 \quad \text{or} \quad x - 3 < -3
$$

Solve:
$$
x > 6 \quad \text{or} \quad x < 0
$$

✔ Answer: $ x < 0 $ or $ x > 6 $

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#### 16) $ \left|\frac{x}{3}\right| + 5 \leq 8 $

Subtract 5:
$$
\left|\frac{x}{3}\right| \leq 3
$$

So:
$$
-3 \leq \frac{x}{3} \leq 3
$$

Multiply all parts by 3:
$$
-9 \leq x \leq 9
$$

✔ Answer: $ -9 \leq x \leq 9 $

---

#### 17) $ |2 + b| - 1 \leq 1 $

Add 1:
$$
|2 + b| \leq 2
$$

So:
$$
-2 \leq 2 + b \leq 2
$$

Subtract 2:
$$
-4 \leq b \leq 0
$$

✔ Answer: $ -4 \leq b \leq 0 $

---

#### 18) $ 5|10 - 6x| - 8 > 92 $

Add 8:
$$
5|10 - 6x| > 100
$$

Divide by 5:
$$
|10 - 6x| > 20
$$

Now split:
$$
10 - 6x > 20 \quad \text{or} \quad 10 - 6x < -20
$$

First:
$$
-6x > 10 \Rightarrow x < -\frac{5}{3}
$$

Second:
$$
-6x < -30 \Rightarrow x > 5
$$

✔ Answer: $ x < -\frac{5}{3} $ or $ x > 5 $

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✔ Final Answers Summary:

Equations:
1. $ r = -4 $
2. $ a = -6 $
3. $ a = -2 $
4. $ n = 0 $
5. $ n = 8 $ or $ n = -8 $
6. $ v = 8 $ or $ v = -8 $
7. $ b = 2 $ or $ b = 0 $
8. $ x = 3 $ or $ x = -3 $
9. $ p = \frac{13}{3} $ or $ p = -5 $
10. $ n = 10 $ or $ n = -\frac{58}{7} $

Inequalities:
11. $ x > -7 $
12. $ n \geq 8 $
13. $ n \geq 7 $
14. $ x \leq -5 $
15. $ x < 0 $ or $ x > 6 $
16. $ -9 \leq x \leq 9 $
17. $ -4 \leq b \leq 0 $
18. $ x < -\frac{5}{3} $ or $ x > 5 $

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